Dept of Electrical & Computer Engineering, National University of Singapore, Kent Ridge, Singapore 119260. Abstract-This paper proposes a compact ...
A Compact Equivalent Circuit Model for the SRR Structure in Metamaterials M. F. Wu, F. Y. Meng, and Q. Wu 1.
School of Electronics and Information Technology, Harbin Institute of Technology, Harbin, Heilongjiang, China 150001
J. Wu 2.
China Research Institute of Radio wave Propagation, Xinxiang, Henan, China 102206
L. W. Li 3.
Dept of Electrical & Computer Engineering, National University of Singapore, Kent Ridge, Singapore 119260
Abstract-This paper proposes a compact equivalent circuit
macro performance of SRR arrays or counterparts. While
model to characterize the split ring resonator (SRR) structure in
accurate,
these
field
solvers
are
time
consuming,
metamaterials, based on the modeling approaches of the spiral
computationally intensive and require experience. These field
inductors in MMIC. In the equivalent circuit model, the lumped
solvers also do not provide any insight into the engineering
elements distribute compactly, and can be directly estimated by
trade-offs involved in the design of new geometry. Although
the geometry parameters of the SRR. Numerical simulation
the better field solvers are excellent for verification, they are
results show that the model can simultaneously predict more than
inconvenient at the initial design and optimization stages.
one resonance frequency of the SRR, and the predicted resonance
To meet the requirement, this paper proposes an approach to
frequency is accurate enough. The equivalent circuit model can be
analyze an individual SRR by a compact equivalent circuit
advantageous to probe the root of the negative permeability,
model based on the spiral inductors modeling methods in
especially
SRR-based
MMIC. The equivalent circuit model can facilitate the analysis
metamaterials exhibiting left-handed properties over multiple
and optimization of SRR-based metamaterials, and predict
frequency bands.
more than one resonant frequency, which is very important for
helpful
for
engineers
to
analyze
SRR-based metamaterials exhibiting left-handed properties I.
INTRODUCTION
In 1999, the negative value of effective permeability was
over multiple frequency bands [3]. II.
THE COMPACT EQUIVALENT CIRCUIT MODEL OF SRR
first achieved by using an array of the metallic artificial atoms called Split Rings Resonators (SRRs) [1]. Since then, many
Researches have been shown that an individual SRR can
research works have been conducted after various methods for
realize the electromagnetic performances of SRRs arrays [4],
realizing negative permeability, especially after the negative
so we had to just look into the performances of an individual
refractive index was experimentally verified [2]. SRR has been
SRR, instead of investigating the entire effective behaviors of
approved to be a useful artificial atom for design and
SRRs arrays in this section. Fig.1 shows the geometry of an
fabrication the artificial media. However, there are still some
individual square SRR, which functions as current sheets while
performances to be improved, such as the anisotropy, lossy and
an incident plane wave magnetic field H is penetrating through
bandwidth with the SRR-based metamaterials.
it. Differing from the spiral inductors, there are two splits in the
It is promising to arrive at the aims through probe the micro
out ring and the inner ring of SRR. So the SRR equivalent
electromagnetic effects of SRR or counterpart, and design or
inductors approximately estimated by four side parts inductors,
improve its geometry. However, most of this modeling work
namely L1, L2, L2 and L3, respectively shown in Fig.2,
has centered on the development of field solvers to predict the
according to the square spiral inductors lumped circuit models.
0-7803-9433-X/05/$20.00 ©2005 IEEE.
APMC2005 Proceedings
L2 produced by one side integrated outer and inner current sheet, while L1 and L3 represent the self-inductors produced by an
LD =
K µ0 n 2 L 2 [ln( ) + 0.5 + 0.178 ρ + 0.0146 ρ 2 + ρ 2π 0.5(n − 1) S 2 (n − 1) S 1 W + t )] + 0.178 × − ln( n n W ( ρ n) 2
integrated sheet and a sheet with a split.
K=
(2 L − 2 S ) − D (2 L − 2 S )
(3)
(4)
Hence the equivalent circuit model and its lumped elements distribution can be shown in Fig.3, and several distributed capacitances also exist in the equivalent circuit model, namely C1, C2, C2, and C3 respectively, which can be determined by the following approaches.
Fig. 1. The geometry configuration of an individual square SRR
Fig. 3. The lumped element distribution in equivalent circuit model of SRR Fig. 2. The SRR self-inductors calculation divisions in four sides
The total capacitances with SRR consist of two parts. One
The self-inductors produced by one side integrated outer and
part is the coupling capacitance between the outer and the inner
inner current sheet such as L2 can be directly estimated by
rings, and the other one is produced the electric charges accumulate at the two splits. The coupling capacitance Cc can
followed formulas [5]: LD =
µ0 n L 2 [ln( ) + 0.5 + 0.178 ρ + 0.0146 ρ 2 ρ 2π
be estimated by the following equation [6] [7]:
2
0.5(n − 1) S 2 (n − 1) S 1 W + t )] + + 0.178 − ln( 2 n n W ( ρ n)
ρ=
nW + (n − 1) S L
(1)
Cc = [0.06 + 3.5 × 10−5 (rout + rin )]
(5)
Then the coupling capacitance should be divided into four (2)
Here, t represents the vertical parameter of SRR, and the lateral
equal counterparts named C0 for SRR four sides, hence: 1 C0 = [0.06 + 3.5 × 10−5 (rout + rin )] 4
(6)
parameters are completely specified by the following:
Here rout and rin represent the radius of the outer circumcircle
1. 2. 3. 4.
and the inner circumcircle of SRR respectively.
The number of turns, n The metal width, W
It is vital to estimate the capacitances at the outer and inner
The length of out ring sides, L
splits, but it is difficult because of the intense electromagnetic
The space between inner and outer rings, S
brink effects, so they are estimated by corrected the parallel
However, L1 and L3 cannot be directly estimated by equations (1) and (2), because of the errors caused by splits. So a new variable named integrality coefficient K should be inserted to rectify the self-inductors of one side, thus L1 and L3 can be estimated by following expressions:
plane capacitance formula as the following approaches: Cinner split = 3ε 0 s/D
(7a)
Couter split = 25ε 0 s/D
(7b)
Here s means the effective parallel plane area, and D means the distance of the parallel planes at the splits.
The distributed capacitances C1, C2 and C3 can be determined by equations (6) and (7) as following approaches: C1 = C0 + Couter split C2 = C0 C3 = C0 + Cinner split
(8a) (8b) (8c)
which display the Z21 coefficient phase degree abruptly changes 180 degrees at above three frequency points. All above simulation results indicate that the equivalent circuit becomes resonant at 6GHz, 15.5GHz and 26.5GHz respectively, which are good agreement with numerical results with error less than 8%. Conclusions can be made based on above results that SRR has three resonant frequency 6GHz, 16.8GHz and 27.8GHz
III.
NUMERICAL SIMULATION AND DISCUSSION
respectively, and can be precision-predicted by the equivalent circuit model.
In order to testify the proposed equivalent circuit model, numerical simulations are necessary. Let’s choose a set of
IV.
CONCLUSION
geometry parameters of SRR shown in Fig.1 such as: L=6mm, W=0.5mm, S=1mm, D=1mm, and the vertical size t=0.5mm.
Based on the spiral inductors modeling method in MMIC,
Firstly, CST Microwave Studio is considered to predict the
the proposed equivalent circuit model has compact distributed
resonance frequencies of SRR. Shown in Fig.4, a vacuum
elements and they can be directly estimated by the geometry
waveguide is employed, then its transmission coefficient is
parameters. Numerical simulation results testify that the
nearly 1 and its phase of image part of the Z21 is shown in Fig.5. equivalent circuit model can simultaneously predict three The results indicate that electromagnetic wave Propagate
resonant frequency points with high precision, which is
through the vacuum waveguide with no reflection, and its Z21
extremely important for SRR-based metamaterials exhibiting
coefficient phase degree abruptly changes 180 degrees at about
left-handed properties over multiple frequency bands.
20.5GHz. Then SRR can be centered in the vacuum waveguide shown
ACKNOWLEDGEMENT
in Fig.4, thus the transmission coefficient of the waveguide with SRR is simulated shown in Fig.6, which display S21
This work was partly supported by a grant from the National
abruptly drops to less than –35 dB at 6GHz, 16.8GHz and
Natural Science Foundation of China (No. 60372051), and
27.8GHz respectively, and the Z21 coefficient phase degree
Fund for the National Key Laboratory of Electromagnetic
abruptly changes 180 degrees at above three frequency points
Environment (No.514860303).
besides 20.5GHz, shown in Fig.7. The simulation results indicate that the vacuum waveguide become almost reflective
REFERENCES
at three frequency points because of the SRR centered in it, thus we can estimate that the SRR has its resonance at 6.0 GHz, [1]
J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart,
16.8 GHz and 27.8 GHz respectively.
“Magnetism from conductors and enhanced nonlinear phenomena,”
Finally, we will employ the equivalent circuit model to
IEEE Transactions on Microwave Theory and Techniques, vol. 47, no.
predict the resonance frequencies of the SRR. According to its geometry parameters L, W, S, t and D, we can apply the
11, pp. 2075-2084, November 1999.
[2]
formulas (1)-(8) to determined the distributed elements L1, L2, L3, C1, C2 and C3。Then the transmission coefficient of the
R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77-79, April 2001.
[3]
Hongsheng Chen, Lixin Ran, Jiangtao Huangfu et.al. Metamaterial
equivalent circuit can be obtained and shown in Fig.8, which
exhibiting left-handed properties over multiple frequency bands. J. Appl.
display S21 abruptly drops to less than –35 dB at 6.0 GHz, 15.5
Phys.96, 5338 (2004).
GHz and 26.5 GHz respectively. Furthermore, the phase of its Z21 coefficient can be simultaneity obtained and shown in Fig.9,
[4]
R. W. Ziolkowski, “Design, fabrication, and testing of double negative metamaterials,” IEEE Transactions on Antennas and Propagation, vol.
51, no. 7, pp. 1516-1529, July 2003.
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[7]
Qun Wu, Ming-feng Wu, Fan-yi Meng, Jian Wu, and Le-wei Li.
Sunderarajan S. Mohan, The Design, Modeling and Optimization of
Modeling the Effects of an Individual SRR by Equivalent Circuit
On-Chip Inductor and Transformer Circuits, Ph.D. Thesis, Stanford
Method, 2005 IEEE AP-S International Symposium and USNC / URSI
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Fig. 4. The geometry of vacuum waveguide with
Fig. 5. The phase degree of the Z21 of the vacuum waveguide without SRR
centered SRR
Fig. 6. The transmission coefficient S21 of the vacuum waveguide with centered SRR.
Fig. 7. The phase degree of the Z21 of the vacuum waveguide with centered SRR 200
0
)) 1 ni 150 Z( g a 100 m (i e s a h p 50
-20
)) 1, 2( S ( -40 B d -60
0
-80
0
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
freq, GHz
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
freq, GHz
Fig. 8. The transmission coefficient S21 of the vacuum
Fig. 9. The phase degree of the Z21 of the vacuum
waveguide with centered SRR predicted by the
waveguide with centered SRR predicted by the
equivalent circuit model.
equivalent circuit model
